Sylvester-Gallai theorems depend on the base field.
For instance, it is well known that any nonsingular planar cubic curve over \(\)  has nine inflection points, and that any line passing through two such points passes through a third [1]}.
These nine points are not collinear, and therefore form a counterexample to the Sylvester-Gallai theorem when the underlying field is changed from \(\)  to \(\) .
In 1966, Serre asked if there are configuration of points in \(^{n}\)  that satisfy the Sylvester-Gallai that are not coplanar [2]}. In 1986, Kelly noted that no such configurations can exist, namely, that points in \(^{n}\)  that satisfy the Sylvester-Gallai property must be coplanar [3]}. Kelly showed that the planarity of Sylvester-Gallai configurations is a simple consequence of Hirzebruch's work on line arrangements which relies on deep results from algebraic geometry [4]}.
